Observation of Non-Markovian Spin Dynamics in a Jaynes-Cummings-Hubbard Model Using a Trapped-Ion Quantum Simulator

The Jaynes-Cummings-Hubbard (JCH) model is a fundamental many-body model for light-matter interaction. As a leading platform for quantum simulation, the trapped ion system has realized the JCH model for two to three ions. Here, we report the quantum simulation of the JCH model using up to 32 ions. We verify the simulation results even for large ion numbers by engineering low excitations and thus low effective dimensions; then we extend to 32 excitations for an effective dimension of $2^{77}$, which is difficult for classical computers. By regarding the phonon modes as baths, we explore Markovian or non-Markovian spin dynamics in different parameter regimes of the JCH model, similar to quantum emitters in a structured photonic environment. We further examine the dependence of the non-Markovian dynamics on the effective Hilbert space dimension. Our Letter demonstrates the trapped ion system as a powerful quantum simulator for many-body physics and open quantum systems.

Figure 1

Experimental scheme. Two counterpropagating global Raman laser beams are applied perpendicular to a chain of 2 to 32 ions to generate a Jaynes-Cummings-Hubbard (JCH) model Hamiltonian. The beat frequency of the Raman laser beams is tuned close to the red motional sideband, and is stabilized by a phase-locked loop (PLL). The internal state of each ion is coupled with its local oscillation by the Raman laser beams as a Jaynes-Cummings model, and these local oscillation modes of individual ions are further coupled by their Coulomb interaction.

Figure 2

Comparison between experimental spin dynamics and theoretical prediction for small Hilbert space dimensions. Blue dots (with error bars representing one standard deviation) are experimentally measured $⟨{\sigma }_z^i(t)⟩$ for individual ions, and the red curves are the corresponding theoretical results by exact diagonalization with no fitting parameters. (a) Two ions initialized in $|\uparrow ,0{⟩}^{\otimes 2}$ (two total excitations) under $\mathrm{\Delta }=-2\pi \times 60\mathrm{kHz}$ and $g=2\pi \times 11.6\mathrm{kHz}$. The evolutions of the two ions are symmetric so only one is plotted. (b) Similar plot for a central ion (ion 4, labeled by 1 to $N$ from left to right) in an $N=8$ chain under $\mathrm{\Delta }=-2\pi \times 60\mathrm{kHz}$ and $g=2\pi \times 11.5\mathrm{kHz}$ with the initial state $|\uparrow ,0{⟩}^{\otimes 8}$ (8 excitations). (c)–(e) $N=20$ ions with two total excitations initialized in $|\uparrow ,0{⟩}^{\otimes 2}\otimes |\downarrow ,0{⟩}^{\otimes 18}$ under $\mathrm{\Delta }=-2\pi \times 5\mathrm{kHz}$ and $g=2\pi \times 10\mathrm{kHz}$. (c) Ion 1 and (d) ion 2 are the two ions being excited on the left of the chain, while (e) ion 20 is the ion on the right. (f)–(h) $N=32$ ions with one total excitation initialized in $|\uparrow ,0⟩\otimes |\downarrow ,0{⟩}^{\otimes 31}$ under $\mathrm{\Delta }=-2\pi \times 5\mathrm{kHz}$ and $g=2\pi \times 11.6\mathrm{kHz}$. (f) is for the excited ion 1 on the left, (g) for its neighbor ion 2, and (h) for the ion 32 on the other end of the chain.

Figure 3

Markovian and non-Markovian spin dynamics in different regimes of the JCH model. We initialize the system at $|\uparrow ,0{⟩}^{\otimes N}$ with $M=N$ total excitations. We fix $g=2\pi \times 12.9\mathrm{kHz}$ and tune $\mathrm{\Delta }$ to observe different spin dynamics $⟨{\sigma }_z^i(t)⟩$. (a) Experimental results for $N=4$ ions with $\mathrm{\Delta }$ ranging from $-2\pi \times 120\mathrm{kHz}$ to $2\pi \times 60\mathrm{kHz}$. The response strongly depends on the location of $\mathrm{\Delta }$ with respect to the phonon band from $-2\pi \times 52\mathrm{kHz}$ to zero. (b)–(g) Dynamics in different parameter regimes in (a) as indicated by black horizontal arrows for the central (left panel) and edge (right panel) ions. Red curves are the theoretical results from exact diagonalization. (h),(i) Measured spin dynamics for a central ion of an $N=20$ chain with $\mathrm{\Delta }=2\pi \times 30\mathrm{kHz}$ (outside band) and $-2\pi \times 5\mathrm{kHz}$ (near band edge), respectively. (j),(k) Similar plots for an $N=32$ chain with $\mathrm{\Delta }=2\pi \times 30\mathrm{kHz}$ and $-2\pi \times 5\mathrm{kHz}$, respectively.

Figure 4

Collapse and revival in spin dynamics under various Hilbert space dimensions. (a) and (b) Experimental data (blue dots with error bars for one standard deviation) and theoretical results (red curves) for an initially excited ion in an $N=4$ chain with (a) one and (b) two total excitations, respectively. Here, we choose $\mathrm{\Delta }=-2\pi \times 10\mathrm{kHz}$ and $g=2\pi \times 11.5\mathrm{kHz}$. (c)–(f) Similar plots for (c) and (d) $N=20$ ions under $\mathrm{\Delta }=-2\pi \times 5\mathrm{kHz}$ and $g=2\pi \times 10\mathrm{kHz}$, and (e),(f) $N=32$ ions under $\mathrm{\Delta }=-2\pi \times 5\mathrm{kHz}$ and $g=2\pi \times 11.6\mathrm{kHz}$. The system has (c),(e) one or (d),(f) two total excitations. (g) The average spin dynamics ${\sum }_i⟨{\sigma }_z^i(t)⟩/N$ under the parameters of (a),(b) with all the spins initialized in $|\uparrow ⟩$. (h) The average spin dynamics for $N=4$, 20, 32 ions with $M=N$ total excitations. The $N=4$ case is the same as (g). For $N=20$ and $N=32$ ions with $N$ excitations, numerical calculation using exact diagonalization is intractable. Hence, in this case the theoretical curves are not plotted, and the data points are directly connected to guide the eyes.